DYNAMIC BEHAVIOR OF SHORTEST PATH ROUTING ALGORITHMS FOR COMMUN -ETC(U)JUN 80 D P BERTSEKAS N0001-75-C-1183

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00 OSP No. 82933

DYNAMIC BEHAVIOR OF SHORTEST PATH ROUTING

ALGORITHMS FOR COMMUNICATION NETWORKS

DTIC.;.,, ,.. ,,.,e C L - %'T E !Dimitri F. Setekas)

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l €Laboratory for Information and Decision SystemsC.. . MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139

80 84 34

June, 1980 LIDS-TH-1005

Dynamic Behavior of Shortest Path Routing

Algorithms for Communication Networks,

by

I.)Dimitri P./Bertsekas

Department of Electrical Engineering and Computer Science

Laboratory for Information and Decision Systems

Massachusetts Institute of Techqology

Cambridge, Mass. 02139

//

/ - i

Abstract

Several proposed routing algorithms for store and forward communication

networks, including one currently in operation in the ARPANET, route

messages along shortest paths computed by using some set of link lengths.

When these lengths depend on current traffic conditions as they must in an

adaptive algorithm, dynamic behavior questions such as stability, convergence,

and speed of convergence are of interest. This paper is the first attempt

to analyze systematically these issues. It is shown that minimum queuing

delay path algorithms tend to exhibit violent oscillatory behavior in the

absence of a damping mechanism. The oscillations can be damped by means

of several types of schemes two of which are analyzed in this paper. In

the first scheme a constant bias is added to the queuing delay thereby

providing a preference towards paths with small number of links. In the

second scheme the effects of several past routings are averaged as for

example when the link lengths are computed and communicated asynchronously

throughout the network.

-!7

2

1. Introduction

A central operational problem of a communication network involves the

choice of routes used by messages to travel from origin to destination. It

is possible, of course, to choose a fixed route for each origin-destination

pair, but this precludes the possibility of adjusting routes to alleviate

congestion due to variations in average traffic conditions. For this

reason attention has focused on adaptive routi ig strategies

whereby congestion in the network is continuously monitored and routes between

origin-destination pairs are modified in real time so as to keep average

delay per message at a reasonable level. A routing scheme of this type was

implemented in the ARPANET in 1969 and attracted considerable attention.

The main idea in this scheme is to compute in real time an estimate of the

minimum average delay per message for each origin-destination pair and to

route messages along the current minimum estimated delay path. When this

scheme was first implemented, it was noticed that it is

pione to severe oscillations. This behaviour is due to the fact that

delay estimates used to choose routes are themselves affected by the route

choice with a feedback effect resulting. To remedy this situation it was

decided on heuristic grounds to introduce an additive factor, called bias,

to the estimated delay of each link, thereby building into the algorithm

a preference towards paths with small number of hops to the destination

[5] - [7]. This had a stabilizing effect albeit at the expense of considerable

loss of sensitivity to traffic congestion.

The implementation of the minimum delay path idea in the original

ARPANET algorithm had a number of flaws allowing, for example, the formation

of loops. For thj! reason alternative schemes based on the same idea were

3

studied, and a new algorithm called SPF has been developed and imple-

mented [11, [4], [11). The present paper is an outgrowth of the

author's participation in the design study of this algorithm during the summer

of 1978 at BBN, Inc. However, our analysis does not focus on the ARPANET

and the SPF algorithm in particular, but rather is geared towards understanding

the effect of feedback and the nature of the dynamic behaviour of shortest

path algorithms where link lengths depend on current traffic conditions.

We note that the algorithms of this paper are far from optimal since they

are single path algorithms in the sense that at any given time there is only

one path per origin-destination pair along which messages can travel. Better

performance can be achieved by allowing multiple paths as for example in the

optimization algorithm of Gallager 19] or its second derivative versions [2].

On the other hand the hardware limitations of some of the presently existing

networks including the ARPANET preclude the use of such more sophisticated

algorithms. Furthermore, we feel that the mere fact that the algorithm has

been successfully implemented in a network as interesting and influential

as the ARPANET makes it worthy of analysis and investigation. This is

reinforced by the fact that the behavior exhibited by the algorithm is

quite interesting and can ,se nontrivial design problems.

The paper is organized as follows:

In Section 2 we provide a deterministic finite state Markov chain

framework for studying a simple version of the algorithm. We show that for

ring networks the algorithm may tend to oscillate between poor routing paths

and become itself a major contributor to congestion. We also demonstrate

how the use of a bias factor can provide a mechanism for damping oscillations

as confirmed by experience with the original ARPANET algorithm.

4

The finite state model does not lend itself to analysis of more sophisticated

routing schemes and more general network topologies. We consequently introauce

in Section 3 a model of a ring network with a continuum of nodes and a single

destination. This allows us to employ techniques of stability analysis of

discrete-time systems with continuous state space, and enables us to further

quantify the relationship between choice of link lengths and algorithmic

behavior.

The analysis of Section 3 focuses primarily on the effect of using a

bias factor as a damping mechanism. In Section 4 we show that oscillations

can also be damped effectively by making the link lengths dependent on

several preceding routing paths via some averaging mechanism such as a fading

memory scheme or asynchronous link length updating. To our knowledge the

fact that averaging can provide a damping mechanism in a shortest path algorithm

has not been noticed earlier and in fact when we originally approached this

problem at BBN, Inc. there was considerable concern regarding its effect on

algorithmic behavior. It is now believed that the significant degree of

averaging inherently present in the SPF algorithm is in large measure responsible

for the stable dynamic behavior observed in experiments conducted thus far

(ll].

The analysis of Sections 2-4 focuses on ring networks. The ring

topology is central for the extension of our earlier results to more complex

network topologies. This extension is carried out in Section 5 under the

assumption that an equilibrium routing exists. However, by contrast with

ring networks, an equilibrium routing need not always exist for more complex

topologies. We demonstrate via example the mechanism by which such a

phenomenon can occur.

The results and analysis of the present paper can be generalized to

the case where there are more than one destinations. This analysis is

5

straightforward but considerably more complex technically and may be found in (3].

The continuous node model of Sections 3-5 may be criticized on the grounds

that it is unrealistic. On the other hand it is very difficult to provide an

extensive analysis of a more realistic finite node network model. In particular,

it appears impossible to demonstrate the effect of averaging in such a context.

Furthermore we believe that the realism of any algorithmic model must be judged

on the basis of the validity of the conclusions it provides regarding the

behavior of the related practical algorithm. These conclusions in our case

have been verified by extensive numerical experiments with finite node net-

works [4], [3]. In particular the validity of our qualitative results

regarding the role of a bias factor and averaging as damping mechanisms

have been amply demonstrated.

2. A Finite State Markov Chain Model

Consider a communication network with nodes denoted by 1,2,..., N and

directed links denoted by (i,Z) where i is the head node and t is the tail

node. We consider the following algorithm for periodically updating paths

for routing messages.

(A) At the beginning of every time period a nonegative length Dit of

every link (i,k) becomes availabe to each node. Based on these lengths each

node computes a shortest path to each destination and routes messages over

that path during the period.

The standing assumption for algorithm (A) is that the lengths Dit used

in computation of a new shortest path depend exclusively on one or more

preceding shortest paths. This dependence is deterministic via a rule that

for the moment we leave unspecified. As an example D may represent some

6

measure of average delay per message on link (i,k) during one or more preceding

periods perhaps with an added bias factor -a scheme currently implemented in

the ARPANET [1], [11]. By assuming that the dependence of Di on previous

shortest paths is deterministic we also implicitly assume that the input traffic

originating at each node is a stationary stochastic process whose ensemble para-

meters can be adequately measured by time averages. This assumption is not

valid, of course, in practice but is a reasonable approximation to the situation

where the time constant of traffic statistic variations is large relative to

the shortest path updating period (a quasistatic assumption, cf. [9]).

Consider first algorithm (A) applied to a given network for the case where

the lengths Dit depend exclusively on the preceding shortest path. Assume

also that the shortest path algorithm has a fixed rule for breaking ties

between equidistant paths. Then each shortest path uniquely determines the

next shortest path. There is a finite number of possible shortest paths

(also referred to as routings) which we denote by R, R2 ...... RM where M

is some integer. To any initial routing say Ri , there corresponds a1

unique sequence of subsequent routings R. , Ri.... Thus eventually some

routing will be repeated (say R. = R. ), and once this happens the routingIk k+n

sequence will become periodic. Thus starting at R. the algorithm will0

eventually end up cycling through Ri ,...R. k+nl. Of course it is possible

that Ri itself is part of the cycle (k=O), and that the cycle consists of0

a single routing (n=l) in which case the algorithm stabilizes at that

routing.

The model just described is one of a deterministric finite state Markov

chain with states Rl,...RM. From Markov chain theory or by elementary

reasoning it follows that the set of all routings {R ,... R M can be parti-

1 14

7

tioned into a collection of cycles (or ergodic classes), and a collection of

transient routings. If the initial routing is transient it is never repeated

by the algorithm, and if it is part of a cycle the algorithm returns to it

periodically. More than one cycles may exist. Furthermore, each transient

routing leads to a unique cycle.

When the lengths Dij depend on a fixed number (say m) of preceding routings,

a finite state model for the algorithm can be similarly constructed whereby

the state space of the model is the set of all m-tuples of routings. Similarly

the state space can be partitioned into cycles and transient states. Analysis

of such a model is naturally more difficult in view of the increased size

of the state space, and this is more so if D. depends on all precedinS

routings in which case a countable state Markov chain model is necessary.

In what follows in this section we will restrict attention to the case

of a ring network with N nodes shown in Figure 1. Node N is the only

destination and all links are bidirectional. By reversing the directions

of flow and the role of origins and destinations the subsequent model can

be converted to one with a single origin and many destinations. The

traffic input originating

N rN-2

N-2

ri-1 ri ri 1

8

at node i and destined for N is denoted by r.. The routing R. , i = 1,...,Ni, ~ 1

is the one for which all nodes j < i route their traffic in the clockwise

direction and all nodes j > i route their traffic in the counterclockwise

direction

0

ROUTING R1

Figure 2

as shown in Figure 2. Given a routing Ri, the flows on each undirected

link (j-l,j) in the clockwise and counterclockwise direction are denoted

by f (i) and f+(i) respectively and are given by

f~i)i .I 0 if i a j

ri + ri 2 + ... +r if j < i

- ri+r i +1 + ' ' ' + rj 'l if i ,< j i

0 if isi.

We will consider the case where the length D i of a link (i,j) is given

by an equation of the form

(1) Dit = d(fix)

where fi£ is the flow on link(ij) during the preceding period and

d is a real valued, continuously differentiable and monotonically

increasing function of flow with d(O) > 0. For simplicity we assume that

the function d is the same for all links but this does not affect materially

the analysis that follows. Since the flow fit depends only on the preceding

routing the same is true for the length Di . It appears that this simplest

of all possible situations is the only one that can be analyzed effectively

in a finite node network context. The practical situation where D is

taken to be the average time delay for a message to traverse link (i,i) can

be reasonably modelled by a function d of the form

'1(2) d(f P +T + (f

where

P. = Average processing plus propagation delay per messageit

T i = Average transmission delay per message

Qix (f i)= Average queuing delay per message when the average

flow on link (i,k) is f

The quantities P and T are independent of the flow f while the

dependence of Q it on f is determined by the statistics of the traffic

arriving at i and routed through Z. If these statistics can be adequately

modelled by an M/M/l queue then QiZ takes the form [6], [7)

f(f(3) Qi (fi£i i t c i t f i k

10

where C is the transmission capacity of link (i,k). We mention, however,

that on the basis of experiments conducted thus far it is unclear whether

the average delay per message in the ARPANET can indeed by modelled as in

(2). This may be due to peculiarities of the ARPANET hardware which are

little understood at present. We now define the shortest path algorithm

Given a routing Ri we define the distances D (i,j), D +(i,j) of node j to

the destination in the counterclockwise and clockwise directions respectively

by

D (i,j) = d[f9(i)]k=i

I ND (i,j) = dtf(i)J.

k =j+l k

+ iIf D-(i,i) = D (i,i) then the algorithm sets the next routing to R.. If

D- (i,i) / D+ (i,i) the algorithm sets the next routing to R where the node nn

is such that

D-(i,j) > D+(i,j) for j > n

D (i,j) < D (ij) for j < n

It can be easily shown that if D-(i,i) D+ (i,i) the next routing R isn

uniquely determined by the relations above. Given an initial routing R0

we consider the sequence of successive routings Rl 2 k .kl ,

generated by the algorithm.

The quantity d(0) may be viewed as a bias factor. It represents link

length at zero flow. The following proposition shows that if d(0) = 0 and

• 0 R1the first two routings are different, i.e. R 0 R then the algorithm ends

up oscillating between the two extreme routings R1 and RN which is the worst

possible behavior that can occur. In the context of (2) the case d(0) = 0

corresponds to the situation where the processing and transmission delays

Pit and Tit are negligible relative to the queuing delay Qit.

rr11

0 1Proposition 1: Let d(O) =0 and assume that R 0R Then there exists an

index k such that for all k-ek either Rk R and Rk+l =RN k =Rand

k+lR = R1 .

Proof: Let Ri be a routing and assume that the routing subsequent to Ri is

R with n # i. For concreteness assume that n < i. We will show that eithern

i = N or else the routing subsequent to R is R. with j i.

If i #N then since Rn is the routing subsequent to Ri we have

(4) D-(i,n-l) < D+(i,n-l) = D +(i,i).

We also have

(5) D (i,i) D+(n,i),

(6) D (n,i) D; D (i,n-l).

From (4) - (6) we have

D+(n,i) e D(i,i) -; D-(i,n-l) > D-(n,i)

so finally

D+(n,i) > D-(n,i).

It follows that in the routing Rj which is subsequent to Rn, node i will

switch his traffic to the clockwise direction so that j > i.

We can show using a very similar argument that if n > i then

either iwl or else the routing subsequent to Rn is R with j < i.

Thus we have that the number of nodes that lie between two

successive routings is increasing at each iteration if none of these

routings is R or RN . On the other hand if the current routing is R1 or

RN then the next routing will clearly be RN or RI respectively. This proves

12

the proposition. Q.E.D.

0 1Notice that, if d(O) 0, the situation R -R can only occur if

D'(ii) -D +(ii) where i is the node for which R0 =R i Thus if we add any

> 0 to any one of the node inputs we will have R # R 1 and the algorithm

will again end up oscillating between R1 and RN. We provide an example

illustrating the result of Proposition 1. Several additional examples

involving more general topologies and multiple destinations may be found

in [41.

Example: Consider a 16-node ring network where node 16 is the destination.

Letr. = 1 for i = 1,... ,7,9,...,15 and r8 = C > 0. If £ = 0 and the initial

routing is R. then by symmetry all subsequent routings equal R.. If C

is very small but positive then for the case where

d(f) = f

the sequence of generated routings is R8 , R1 0 , R3 , R16 , Rl, R1 6 , RI,...

This fact can be verified via a straightforward calculation in Figure 3

which shows the flow patterns corresponding to successive routings.

We now turn our attention to various notions of equilibria and

stability. We say that Ri is an equilibrium routing if

D-(i,i-1) < D+(i,i-1), and D +(i,i) < D-(i,i).

It follows from this definition that R. is an equilibrium routing if and1

only if it repeats itself via the shortest path algorithm.

We say that a node i is an equilibrium node if

D-(i,i) < D+(i,i), and D +(i+l,i) < D-(i+l,i)

In words a node i is an equilibrium node if he switches his traffic in

both cases where the routing is Ri and R.

_ _ A

Rs R16 L

0A 7 0 + 1+

O 4 O/A 3 E

00 0~

1st Ruting4th Routing

2nd Routing 5hRuIn

RIO1+

5 5 Fiur 0 t otn

14

We say that an equilibrium routing R. is locally stable if routing1

R generates either Ri or R. through the algorithm, and routing R

generates either Ri or Ri+ 1 . We say that an equilibrium node i is locally

stable if routing Ri generates Ri+ 1 via the algorithm, and routing Ri_ 1

generates R.. The definition of local stability is based on the idea that

when the algorithm starts "close enough to equilibrium" it should not lead to

a "growing" oscillation. The following proposition complements Proposition 1

and suggests that the bias level d(O) should exceed a certain positive value in

order for an equilibrium routing or node to be locally stable.

Proposition 2: a) An equilibrium routing Ri is locally stable if

r -1 N-1 r. N-1 -d(O) > max { 2 m£ , m

where

M = max{d'(f)If(i-l) < f < f, (i)} for 2 = 1.... ,i-l

= max{d' (f f+- (i) < f < f+ (i-l)} for = il..... N-1

= max{d'(f)If (i) < f < fi(i+l)} for Z = 1....,-2

i2 + = max{d'(f)lf+(i+l) < f < f+ (i)} for £= i,...,N-1

where d' (f) denotes the first derivative of d at f.

b) An equilibrium node i locally stable if

r. N-1d (0)> - E

-2 91-1

where

M = max{d'(f)lf2.(i) < f < f- . i~l)1 for Z =

m - max d'(f)lf+(i+l) < f < f+(i)} for I - i+l,...,N-l.

15

The proof of Proposition 2 involves a straightforward but lengthy

argument and will be ommitted. It can be found in [4].

Proposition 2 implies that in order to ensure local stability the bias

d(O) should exceed a level that depends strongly on the traffic conditions.

This level is proportional to the input at or near the equilibrium and to a

global measure of the derivative d' along the ring. Thus it may be necessary

to choose a value of d(O) which is large relative to r and d' in order to

ensure stability for a broad range of input traffic conditions. This can be

accomplished by adding a large constant to d. On the other hand this would

introduce a tendency in the algorithm to generate routings close to the min-

hop routing (i.e. one that selects route- according to minimum number of links

to the destination). As a result the algorithm would tend to be insensitive

to congestion. This tradeoff will be reencountered in the next section

The point of view that has been adopted in this section is one whereby

the algorithm is viewed as a dynamic system with a finite number of states

(the finite collection of possible routings). Unfortunately the study of

dynamic behavior and stability properties of such systems is notoriously

difficult. To begin with there is no accepted definition of equilibrium, and

in fact we saw that in the ring network context there are two types of

"equilibria" that are of interest - equilibrium routings and equilibrium nodes.

Furthermore there are no established methodological tools that can be helpful

in a finite state system framework. As a result our progress has been limited

to the results just discussed. We are thus motivated to consider approximation

of the discrete system with a continuous system having a continuum of states.

For such systems there is an effective and well developed stability theory

that can be utilized for analysis. We take this approach in the following

two sections where we introduce a network with a continuum of nodes. Despite

16

the radical nature of thissep the analysis provides informative results

and clarifies the role of averaging the effects of several past routings as

a means of damping oscillatory behavior. The validity of our approach is supported

by the fact that qualitative conclusions drawn from the continuous node model

have been verified computationally in finite node models.

3. A Continuous Model of a Ring Network

We consider a continuum of nodes arranged in a ring and sending traffic

to a single destination as shown in Figure 4.

1/4 3/4

t/2

Figure 4

Points on the ring are identified with their distance t from the destination

in the counterclockwise direction, where t is normalized to take values in

the interval [0,11. Traffic can move on the ring in both directions.

For every t in (0,1] we denote by r(t) the input density at t. The

meaning of the function r is that for any subinterval [tl,t 2 of (0,11 the

total input traffic originating at nodes in it1,t21 is

17

t2r (t) dt

t1

We assume that r is continuous on [0,11 and r(t) > 0 for at least on te (0,i)

Note that a network with a finite number of nodes can be modelled by a function

r containing impulses and such a function can be approximated by a continuous

function consisting of narrow triangular pulses of finite height. We are interested

in routings specified by points y in [0,1], where the flow splits, i.e.points

larger than y send their flow counterclockwise (or in the positive direction) and

points smaller than y send their flow clockwise (or in the negative direction).

To a given function r and routing y, there corresponds at every point t a flow+

in the Positive direction f (y.t), and a flow in the negative f'(yt) given by

t r(T)dT if y< t

(7) f+(y,t) =

0 if t<y

_ 0 if yf t

(a) f (y,t) -

SY r(T)dT if t < yt "

In order to introduce an algorithm such as (A) in the framework of

the continuous model we consider a function d mapping flows into the non-

negative real numbers. The meaning of d is that given a routing y and any

point t, the distances D- and D+ from t to the destination in the negative

and positive direction are given by

t

(9) D(yt) - df(y, ") ] d0

(10) D +(y,t) -~ d[f +(y,T))dr.t

We will assume that d is a monotonically increasing function of f with every-

where continuous derivative. We further assume that d(O) > 0. As Proposition 1

shows, the case where d(O) = 0 is not interesting from a practical point of

view.

We consider the following algorithm for generating routing sequences

{yk}:

(Al) Given a routing yk' the next routing Yk+l is the solution of

the equation

(11) D (yk, Yk+l = D+ (yk yk+l)

It will be shown as part of Proposition 3 that equation (11) has a

unique solution for every ykE[O,l]. Note that since we have

D (yk,t) < D(Ykyk+l) = D +(ykyk+l )< D+(ykt) if t < yk+l

and- + D(k

D (yk,t) > D (yk'yk+l) = D (yk'Yk+l) > D+(ykt) if t > Yk+l

it follows that a routing yk+l determined from (11) is such that every point t

routes its flow in the positive or negative direction according as

D (yklt) > D (yk,t) or D(yklt) < D+ (ykt), i.e. according to minimum distance

to the destination.

We say that y* £[O,l] is an equilibrium if

(12) D"(y*,y*) = D+(y*,y*)

We first show some preliminary results relating to existence and

optimality properties of equilibria:

Proposition 3: There exists a unique equilibrium y*E (0,1). Furthermore

equation (11) has a unique solution yk+l for every yk"

19

Proof: Using (9) and (10) we have for all y and t

(13) oD = d[f (yt)], 6D -d[f(y,t)].at a t"

We have d(O) >0 and d is monotonically increasing,so 6D (y,t) > 0 and

OD+(Y't) < 0. Thus for fixed y, the function D-(y,') is continuous,

monotonically increasing and satisfies D (y,O)=0, while the function

D+(y,.) is continuous monotonically decreasing and satisfies D +(y,l) =0.

Hence the equation D-(y,t) =D +(y,t) has a unique solution in t lying within

(0,1). Denote by g(y) the solution corresponding to y. The function

g:[0,1]-[O,l] can be easily shown to be continuous and, by Brower's fixed

point theorem ([8], p. 161), g has a fixed point y . This y is an

equilibrium. If there exist two equilibria yl and Y2 with yl < y 2' then

since d(f)> 0 for all f> 0, we must have

D-(y*,y* ) < D (yl,Y2) <D-(y2,Y2) =D (y2,Y2)

*+ +( * *D (yY) <D (y2 ,y*) <D lY ) *D (yll

which is impossible. Hence the equilibrium is unique. Q.E.D.

Proposition 4: The equilibrium minimizes over all y C [0,1] the expression

II I+J(y) = I> 1)f(y,t) ]dt f p[f- (y,t) ]dt

where p is any function satisfying for all f

(14) ' (f) d (f)

and p' denotes the first derivative of p.

Proof: The first derivativw J'(y) of J is given by

1 ;if (y,t) +Jll, , -t) -j~

(15) J,(Y) - I f +(y,t)]-f-- - (it + I f (y,t)] Df (y,t) dt. ly:01

20

It can be seen from (7) and (8) that

af + (y.t) -r(y) if y < t i

(16) y 0 if t < y

f 0 if y <t

(17)

r (y) i f t<'y

Combining equations (14) - (17) we obtain

j' (y) ffir(y)[ - fld[f+(y,t)Jdt+ 'od[f-(Y't)]dt]

or equivalently

J'(Y)- r(y)[D(y,y) -D +(y,y)].

If y is an equilibrium it can be seendiat we have

D (yy) S D+(yy) if y 'y

D (y,y) _D+(yy) if y > y

Thus J'(y) <0 if y < y*, J'(y) > 0 if y* < y, and J'(y*) = 0.

It follows that y* minimizes J. Q.E.D.

Proposition 4 shows that one can minimize the integral of

average delay over the ring by choosing the function d to be marginal

delay and by guaranteeing that the algorithm converges to an equilibrium.

The needto use marginal delays as link lenqths in order to minimize total

average delay has been pointed out earlier in a different algorithmic

context [9]. The following discussion, however, casts doubt as to whether

the algorithm will converge to an equilibrium when the link lengths are

21

chosen to be the marginal delays. In any case Proposition 4 suggests that

convergence of the algorithm to an equilibrium is desirable since a function

p satisfying (14) is monotonically increasing and convex and hence an

equilibrium will at least be a reasonably good routing even if it is sub-

optimal in terms of a particular design objective.

we now consider the convergence properties of the algorithm. For

any yc[0,1] we denote by g(y) the unique solution in t of the equation

D-(y,t) = D +(y,t) (c.f.Proposition 1). Thus Algorithm (Al) can be written 4(18) Yk+l = g (Yk)

We have for all yE[0,1]

g(y)

(19) D [y,g(y)] d[f-(y,t)]dt d[f (y,t)ldt = D+[y,g(y)]

dgy)

We evaluate the first derivative g'(y) d y)for yS(0,1). Differentiationdy

in (19) yields

g(y)

f d~f (y,t)]dt +- d[f (y,g(y))]g'(y)0

or = g~y)d'(f (y,t)]dt - dif (y,g(y))]g' (y)

fd'[f+(yt)]dt -jq d'[f (y,t)]dt

(20) g' (y) Z ____________

d[f-(y,g(y))] + d[f+(y,g(y)))]

We have for t # y

(21) 3d(f+(y,t)] = d,+ff+(y,t)) f (y,t)

-(21) af- (y,t)

(22) ad[f (y,t)] = d'[f (y,t)] - y(

3y

22

combining (20) - (22) with (16), (17) we obtain

min{y,g (y) } - +(23) ( y) [ d[f-(y,t)]dt + 4max{yg(y} d'[f+(y,t)ldt]

d[f-(y,g(y))] + d[f+(y,g(y))]

+at the equilibrium y* we have y* = g(y*) and f (y*,y*) = f (y*,y*) = 0, so

(23) yields

r(y*)[ J'df(Y*,t)Idt + f d'ff+(y*,t)]dt(24) g' (y*) = -:

2d(0)

By using a theorem of Ostrowski ([8], pp. 300-301) we can state the following

local convergence and rate of convergence result for algorithm (Al).

Proposition 5: Let y* be the equilibrium. Then if Ig' (Y*)f < 1 or

equivalently

() d r(y*)[ dhf(y*,t)]dt +J*d'[f+(y*,t)]dt](25) d (0) > I

2

there exists an open interval I containing y* such that if y0 eI the sequence0P

{ykI generated by algorithm (Al) remains in I and converges to y*. Further-

more if yk 3 y * for all k there holds

k1

(26) lira sup _Yk+l I = lim sup ly - y* I = g'(Y*)l(26)y li T y217 kim k 1

When the equilibrium y* has the property specified in the first conclusion

of Proposition 5 we say that it is locally stable. If Ig' (y*) > 1 then the

linearized system corresponding to yk+l = g(yk ) is unstable, so the

algorithm tends to diverge from y* when started close to it. Notice the

similarity of equation (25) with the corresponding local stability conditions

for finite node networks (cf. Proposition 2).

23

A sufficient condition for global convergence of algorithm (Al)

can be obtained by requiring that g be a contraction mapping, i.e. for some

pE(O,I) there holds

(27) Ig() -y * l : y-y * I, , VyE[0,1].

From Taylor's theorem and the fact g' (y) < 0 we have

g(Y) - Y*= I '(z)dzl

Let

j=max d' (f)

0O-r f j r (t) dt

0

From (23) we obtain for all z

Ig, (Z) z ~~zf - g (Z)2d(0)

Thus (27) is satisfied if

y__y <I

sup 2d(.) y-yyE[0,l 2

y y

or equivalently ifI r(z) 1 Iz -g(,Z)[dz

(28) d(0)> sup *y.

yE[O,l] y-yy y

24

This will be true in particular if

(29) d(0) > OR2

where R=max r(t). The conclusions of the preceding discussion are0 ' t : i

summarized in the following proposition.

Proposition 6: If condition (28) or the stronger condition (29) holds,

every sequence {y I generated by algorithm (Al) converges to the equilibrium y*.

When the equilibrium y* has the property specified in Proposition 6 we

say that it is globally stable.

In order to put the results obtained thus far in better perspective

let us write d(f) as

d(f) = a + d(f)

where a = d(0) represents the bias factor. For fixed input density r we

have that to each positive value of bias a there corresponds an equilibrium

y The equilibrium is locally stable for a satisfying [cf.(25)]

(30) L > X

2

and globally stable fora satisfying [cf.(29)]

BR

(31) a > R

As 0 increases the corresponding equilibria tend to become stable. Further-

more from (24) and (2?) it can be seen that the speed of convergence of the

algorithm is accelerated as a increases. On the other hand it is easy to see

* 1that y- * as a w , ich in the context of the routing problem means that

L d

25

the algorithm becomes incr4 -singIy ir!vensitive to congestion as 00"o

Since in a practical situation we are interested in the stability Properties

of the algorithm for a broad range of inputs let us consider input densities

of the form

(32) r (t) = Ar(t)

where A is a positive parameter. Then it is clear that as X increases a larger

value of bias is necessary in order to stabilize the algorithm.

For example if d is of the form

(33) d(f) = ' + fn

where 3>0, n>O then from (3) and (31) we see that if r is changed to Ar

nas in (32), then the stabiliLv., threshold level of the bias is multiplied by X

Thus for fixed a and r there is a choice of X for which the corresponding

equilibrium is unstable. Incidentally the expression (33) for d has an

interesting property, namely,that the set of all possible equlibria {yaU>0}

as well as the set of all locally or globally stable equilibria is independent

of the level of input A and depends only on r. This is straightforward to

verify using (33) and the fact that if r is changed to Ar and a is changed

to Ana then the routing sequences generated by the algorithm are unaffected.

Choosing the Bias as a Function of the Current Routing

Since stability of the algorithm depends strongly on the level of bias

and the level of input we are motivated to consider schemes where the bias

is not held fixed but is rather adjusted adaptively on the basis of currently

available information. An interesting scheme is to use a length function

of the form

26

if (Y)+ d(t)

wICJ d 1~i on , i iuousiy 4ifferentiable, monotonically increasing function

with ik0) C, .i-l x(y) i.. taken to be some monotonically nondecreasing

, . f , )ldt + d f (y,t)ldt.

"C'" '--n C> 2 untior. of the form

.,-y) + Y,[D1T(Y)]2

L i,:0 X.:Iiimontally determined nonnegative constants

;or~~~ms1 ku a,<. ni ,ontext of a finite node network with not necessarily

r srotc~ui , sche like this can be very easily implemented. In this

( c . ho caiciubat,. as the sum of all reported link "delays"

S 'iis (y) can ,e computed by each node via a formula such as

(34) and t ,o link lenqt can he computed as Di= (y) + d(fi)-

,; ih , e thc !vi'vp, just described can be analyzed along similar lines

a5 ', ri ir ii. t : 2.;t 2 L. it hac be-en tested in quite extensive numerical

t:xpti>lwInt:. ir,.IJie I finlt. node networks and it was shown to have very

fac~ >o y pcrr -manhc. [4] , [3]. This can be attributed to the fact that

, I. [vcif bia'; incre s or decreases with the level of input thus

'J~i n olitomine ,alinq with respect to input level. In fact it can be

!hia i'Lar it d ha.; the. form d(f) = Bfn where >O, n>O and we choose

,4 y) Y1D,,(y) wherc Y( i... for every input density function of the form

\c t), 4, th. :,'. onerit-i by the algorithm do not depend on X.

If

27

4. Averaging the Effect of Several Routings

In this section we show that the stability properties of the shortest

path algorithm of the preceding section can be improved if link lengths suitably

depend on flows corresponding to several past routings. There are several

possibilities along these lines. Some examples are as follows:

a) Averaging over the present and the past n routings.

Given a sequence of past routings ykYk-l,..., we define for any

t in [0,1] "averaged" distances to 0 and 1 by

0 i=0

(36) +(kyy k-n't n+l i=0 -i

1 n

Thus distances are calculated by integrating -4j i 0 d[f(yk i, T )Iwhich is

an averaged length over the routings yk' ..... k-n' in place of d(f(ykT)]

which is the length corresponding to the last routing.

The new routing yk+ I is obtained from the equation

(3) -ykk_ .... ,k-n, Yk+l ) =D +(ykYk-l,... ,Yk-nYk+l "

It is easily seen that this defines uniquely k in terms of ykYkl,

As earlier we write the corresponding equation as

(8) Yk+I = g(Yk'Yk-I ..... Yk-n ) "

A routing y is said to be an equilibrium if

jL

28

y g(y ,y ,...,Y ).

It is clear that y is an equilibrium in this sense for a given bias level

if and only if it is an oquiii.brimn in the sense given in the preceding

section.

We can define local stability of y in the obvious way. We have

that y is locally stable ii it is also a stable equilibrium of equation

(38) linearized around y (5cc [8] p. 353). It is a known fact that this is

true if all roots of the characteristic polynomial

C P n+l 1 3-L n Og(v) n-iC (p ) p k p - P - ''

k k- Ik-n+l 6Yk-n

lie inside the unit circle, (i.e. have modulus less than unity). We cal-

culate the derivatives

We have for a>0 similarly as earlier for every i

* y1

2,1 ( --+ 'j d'[f-(y*,t)]dt +j d [f (y*,t)]dt}k-i 2d() n+ 0 y

Define

(39) d r' [ ' [f-(y*,t)]dt + J , d,[f +(y,t)]dt}2d(0) "0 y

Note that, from Proposition 5, y is locally stable for algorithm (Al) if

p<l. The characteristic polynomial can be written as

(40) () n+ I + n _ n- I + _'40 '~) + 1-I P i-I p " + n+l n+l

We now use ti,( iollowing fact:

Lemma: Let g be a po!itiv, scalar and n be a positive integer. The roots of

the polynomial

29

n+ n - n-Ip +.p +lp +...+p+;

lie inside the unit circle if and only if >".

Proof: This result can be shown by straightforward application of Jury's

stability test ([10], p. 97-98). Q.E.D.

We now apply the result of the lemma to our problem. We have

that the equilibrium y will be locally stable if

n + I.

It follows using (39) that in the averaged algorithm the bias level must

satisfy{* y 1

r(y If dt + d' f(,.*,t)] dt

d(0) > 0 y2 (n+l)

in order for the corresponding equilibrium y to be stable. if we compare

this with the earlier algorithm [ci . (2:,)] we sve that in the averaged

algorithm the bias threshold level for stability is r duced by the factor

I1 over the one of al,;orithn (Al). For a given traittic input, and anyn+ I

given bias level the corresponding equilibrium can be made stable by

averaging delays over a su'ficiently larpe number (A periods.

Regarding rate ol convergence, 0 trowski's Theorem again applies.

We have from the proof of Th. 10.1.3 of [8] that given any >0 there

exists a norm on R :;uh that if y 11 y for all

(Y~k+1- Y, ... 'kn+lY )

lim sup k; ,*l P(n)4 fk .... yk-Y , .. -y )

where P(l4,n) is the rnayimi:w r,,t modulus oi the biaractcristic polynomial

C(p) of (40). It can be sUeen that for fixed n we have ¢ (i,n) 0 as - 0.

30

If are the roots of C(p) we have p . . =Pn- so thatP l nlth

P(a,n) t% q ) It follows that for fixed L we have P(4,n)-l as n- - ,

so that the rate of convergence deteriorates as n-'. Thus too much damp-

ing can slow down the speed of convergence of thu algorithm.

b) Fading Memory SchemeThis scheme is similar to the preceding one except that the lengths

corresponding to all past routings are averaged via a fading memory scheme. Given

the sequence of all past routings yk'lyk l ... }, the next routing Yk+l is

determined as the solution of the equation

i~ +

(41) j 6( t)dt f C (t)dt

0 Yk+lk

where 6k and 6+ are obtained by the following recursive fading memoryk k

scheme with decay factor E[OI)

6 k(t) = 06k (t) + (I - )d [f-(ykt)J

+ (t) + (1 - )d~f( 0t)]k~t k 6kI [f(k) "

Alternatively we can write

k(42) 6k(t) = (- ) i Dk-id[f-(yilt)]

kk

Let us write the solution of (46) as

(44) Yk+l m g(Yk'Yk-l1 . ... )"

Let us also consider the linear system obtained by formail linearization of*

(44) around the equilibrium y . We have similarly as earlier that this

linearized system is

2S(45) Yk+l f- ( )y + Yk[+ k- + "'']"

31

where p is given by (39). Let us denote

2Zk+l Yk + PYk-I + r Yk-2 +

Then we have for all k

(46) Yk+l = - 40 -")Yk - k( - 0)'z k

(47) Zk+l = Yk + Zk

and it follows that the linearized system (45) is in effect the two-

dimensional system described by (46) and (47). This latter system is stable

if both eigenvalues of the system matrix

lie within the unit circle. These two eigenvalues can be calculated to be

0 and B- (l- ). It follows that the linearized system is stable if

Although we do not provide a proof, it is possible to establish rigorously

that stability of the linearized system (45) implies local stability of

the algorithm (44) and thus we have the result that the threshold value of

bias, for stability in the f: ling memory scheme is reduced by the factor

-L1 over the one of algorithm (Al). The optimal speed of convergence is

obtained when the eigenvalue - (l - 0) equals zero in which case a super-

linear rate of convergence is obtained. This is so when =--- For other

values of in the interval , ) the rate of convergence is linear, andS+

32

for a-< the equilibrium is unstable. As ( is increased from the optimal

value towards unity the rate of convergence deteriorates.l+ji

c) Asynchronous Length Reporting

This type of scheme is patterned after a shortest path routing algorithm

where nodes report asynchronously the lengths of their outgoing links and the

shortest paths are updated after each report. The set of nodes 10,1) is

partitioned into n subsets which we call SIS .... ,Sn At some time, say 0,

the nodes in S1 report their lengths averaged over the flows corresponding to

the preceding n routings and a routing update takes place. Then at time

a > 0 the nodes in S2 do the same thing. Similarly, for i = 1,..., n-l, at

time (01 + a + ... + 0.) the nodes in S do the same thing. At time1 2 ii+l

(a + 2 +...+O n ) the nodes in S again report their lengths, an updating1 2 n1

takes place and the process is repeated. This type of asynchronous operation

is currently in use in the ARPANET [4] where, in a finite node network

context, S. consists of a single node for all i. There are also otheri

variations of asynchronous operation involving for example averaging over

all preceding routings via a fading memory scheme. This type of algorithm

is described and tested computationally in [4] and [3]. The analysis of

all these schemes is very similar as that of the averaging schemes described

earlier in this section. The details are quite messy and may be found in

[4], where it is shown, via analysis and computational experiment, that

asynchronous operation has a .ubstantial beneficial effect on the stability

properties of the short(st path algorithm.

5. The Case of a Network with an Arbitrary Topology

The extension of the continuous model to the case of a network with

33

arbitrary topology is quite straiqhtforward. However, the notation required

for a precise mathematical description is very cumbersome and tends to cloud

the main ideas. For this reason our pr scntation will be somewhat informal.

Consider the case of an undirected network with a single destination.

Let r be the input density function mapping points on the undirectred links

of the network to the nonneqative real numbers. The mcaning of r again is

that, given any interval I on a link, the total traffic input oriqinatinq

at this interval is the inteqral of r over I. We view the set of points on

the network as a subset of a Euclidean space of appropriate dimension, and

assume that r is a continuous function. In order to consider notions of

length we associate with each undirected link (i,P) two directions i- and

k--i. (There may be more than one links connecting a pair of nodes within

our framework. When we refer to a link (i,9) we mean a particular link

connecting i and Z and specify further when there is danger of confusion).

A length function 6 is a function which assigns to each point on an undirected

link (ij) two nonnegative numbers one associated with the direction i*V

and the other associated with tlic2 direction Q i. We assume that is piece-

wise continuous along every link in each direction. The meaning of 6 i's

that given any two points on a link (i,9) their dki tance in the direction 1 *

is obtained by integrating S as defined in that direction between the two,

points. The distance in the opposite direction -i' is definod ana]oou. ]y.

Similarly we can consider path,; between Point; on possibly different links

and define their long th in one or the other irection.

We now associate to a ivon lenqtIh function a shortest path of t-Vi ay

point, and an as!;r-iat,, r' tit irni. W,* ,j!:!7jmf that , is everywher, ;o : ,i ,.

(;ivn any point w-, tin: dr t- i t ion e, pt ; to th' ,lortimat (, n n

34

their associated distances specified by the function 6. A path of minimum

distance is referred to as a shortest path from the point to the destination,

and the corresponding distance is referred to as the shortest distance of

the point to the destination. The routing corresponding to 6 is the set

of points for which there are more than one equidistant paths to the

destination. A routing is said to be regular if it does not contain any

nodes of the network, otherwise it is said to be singular.

Given the function 6 , a shortest path of each point and the corres-

ponding routing can be constructed in a simple manner along similar lines 4as for usual networks. We first construct a shortest path tree for the

network in the usual manner by using as(directed) link lengths those speci-

fled by the length function 6. (The length of the directed link (i,A) is

the Integral of 6 along (i,l) in the direction i-1). This gives us a

shortest path and the associated shortest distance for every point on the

shortest path tree including all the nodes of the network. A shortest path

for points on links that are not part of the shortest path tree can be

obtained as follows:

Let (i,A) be a link that is not on the tree. Let Di and DA be

the shortest distances of nodes i and A. The shortest distance of a point

t on (i,L) is

'iAD(t) -min ID i+ J 6 i()dt , D+ J 61 ()dt}

t t

where 6 is 6 in the direction A i and 6 is 6 in the direction 1 -.

It can be seen that the routing corresponding to 6 is regular if and only

if each (ordinary) node of the network has only one shortest path associated

with it. If a routing is regular then every one of its points Lies in the

t

35

"interior" of some link. Notice that the preceding construction shows that

a routing (regular or not) consists of (L - N + 1) points where L and N are

the number of undirected links and nodes respectively.

Given a shortest path tree and the corresponding routing constructed

as just described, we can define the flow corresponding to it. At each

point, say t, of a link (i,Z) there are two flows to consider (one of which

is zero); the flow in the direction i-* and the flow in the direction -i.

Each is defined in the natural way by integrating the input density function 4r over the portion of the network that lies "upstream" from the point t,

i.e. over the set of points the shortest paths of which meet t on their way

to the destination. At the points of a regular routing the flow is zero in

either direction. Notice that if 6 is such that the corresponding routing

is regular the flow is uniquely determined by 6 . Otherwise the flow will

depend not only on 6 but also on the shortest path tree selected.

Suppose we are given a monotonically increasing, continuously

differentiable function d mapping flow into the positive numbers. Given a

shortest path tree T corresponding to a length function 6 with routing Y we

can define a new length function & which assigns to points t in any one of

the two possible directions the length 6(t) = d[f(t)] where f(t) is the

flow at t corresponding to 6 and T in the appropriate direction. The

corresponding routing is denoted Y. Note that if Y is singular then X and

Y depend not only on S but also on T. If Y is regular then Y is uniquely

determined by (S

We are now in a po,ition to define an algorithm cimi lar to the one of

Section 3. Given a ],,ngth function 6 and a corresponding shortest pathn

tree T and routing Y W the next le'ngth function i ; 7 with Y0 1(-orl'e!i,' inI' Y 1 (

routing Y1 .0 A T:hor test path tre:e T I 001 *,j 'end ng to 1 5 iS 1 ], et (,

36

and is used to define similarly 2' Y 2 and T Similarly the algorithm

generates 6k, Yk and Tk for all k.

We say that a routing Y* corresponding to a length function 6* and

shortest path tree T* is an equilibrium routing if 6 = (" and Y = Y*.

Contrary to the case of a ring network where we were able to prove

existence of an equilibrium, in general there need not exist an equilibrium.

This fact is demonstrated in the following example and provides an indication

of the complexity of the dynamic phenomena that we are investigating.

Example: Consider the network shown in Figure 5. 4

i:

/ i

\\

Figure 5

There are two nodes 1 and 2 and th~ree links connecting them denoted by

A,B,C. Node 2 is the destination. Points on A,B, and C are parameterized

by their Euclidean distance Lo the destination. The Euclidean lengths of

A,B and C are all taken equal to unity. Let the input density function be

as follows

37

For link A: r(t) E 1, VtE[o,1]

For link B: r(t) m rB) VtE[o,i]

For link C: r(t) E rc, VtE[0,1].

We assume that lr B : r l<r Let

d(f) = C' + f

where Y > 0 is the bias factor. K

In view of the fact 1 < rB < rc, l<rc, it is clear that an equilibrium

routing cannot contain a point in the interior of link A, while it must

contain a point in the interior of link C. We consider two cases:

Case 1: r B = 1. Then an equilibrium routing cannot contain a point in the

interior of link B so the only candidate for equilibrium are the two types

of singular routings shown in Figure 6. In routings Y and Y the incoming

traffic at node 1 is routed through link A and link B respectively. None

of the two routings can be an equilibrium. In routing Y1 there will be points

in the interior of link A which will have a shorter distance to the destination

(corresponding to Y1 ) through link B rather than through A, and the reverse

situation occures in routing Y Notice that this argument makes use only

of the magnitude of r and r and is independent of the form of the function d.B C

Case 2: l<r Then it can be seen that the only candidates for equilibria

are routings of the form shown in Figure 7. Each equilibrium routing candidate

38

is specified by the points yB' yce[Ol] where the flow separates on links B and

C. We have that the distances D+(yB), D-(yB) of Y b corresponding to routing

(yBYc) along the counterclockwise and clockwise paths respectively are given

by

D (y B B+rB J 0 (yB-t)dt0

+J [r B(I - Y + r C(1 - + (1 - t)Id t

If (yBPyC) is an equilibrium we must have

D"(yB) = D+(yB)

which after some calculation can be written as

(48) 2(O+rB)(1-yB) +rC( - yC) =B 2

By symmetry the equation D-(y ) =D+(y c) can be written as

rC - 1

(49) rB(l - yB) + 2 ('+ rc) ( -y) C

Equations (48) and (49) are in fact necessary and sufficient conditions for

(yB yC) to be an equilibrium routing. Thus there exists an equilibrium* *:

routing if and only if the solution (yB3yC) to these equations satisfies

yBE[0,11, y*E[0,11. After some calculation, this condition can be shown to

be equivalent to

rC( 2rB - r C I)(50) B - 1)

39

3.i

A B ( A B

Routing YI Routing Y2

Figure E

F//

Figure 7

40

If 2rB > rC + 1 then for every level of bias there exists an equilibrium rout-

ing (yB' YC )" If however 2 rB < rc + 1 then there exists an equilibrium only

for a above the threshold level indicated in (50).

The preceding example shows that existence of an equilibrium can depend

on both the level of bias and the input density function. Furthermore, it

may happen that, for a given input density function, no value of bias can be

found for which an equilibrium exists. This last phenomenon is of a singular

nature and is due to the fact that the Euclidean lengths of links A,B, and C

are all equal to unity. To see this consider the routing Y, corresponding

to the length function 6+ (t) H 1, 6-(t) E 1. The routing Y is analogous

to the min-hop routing in discrete node networks, and can be associated with

infinite level of bias. It is an equilibrium routing for the case d(f) £ 1.

If Y is a regular routing, i.e. each node has a unique minimum Euclidean

distance path to the destination, then it is clear that, for any given input

function r, there exists a threshold level of bias a such that for all

a >cta regular equilibrium routing exists.

Characterizing the dynamic behavior of the algorithm in the absence

of an equilibrium is certainly an interesting problem but we have been un-

able to make much progress in this direction. Computational results for

finite node networks given in [31 suggest that the stability properties of

the algorithm are improved by high level of bias and averaging similarly

as in the presence of an equilibrium. In what follows in this section we

restrict attention to the case where a regular equilibrium routing exists.

Given a regular equilibrium routing Y* = ... ,y* considern

for j = 1,2,.. .,n the link (ij ,£) containing y* and the two shortest paths

from y* to the destination. A simple but fundamental observation is that

3

these two paths join at some point thiereby forming a ring of the type con-

41

sidered in Section 3. The zero point on this ring is the point where the two

paths join. Let e. be the Euclidean length of the ring containing y. For

j = 1,2,..., n we parameterize points on the ring containing yj by the number in

[O,e.] going from smaller to larger numbers as we traverse the ring in a chosen

direction similarly as in the previous two sections. Thus points y. on

the link (ij,.j) can and will be identified by the number in [O,e.] specifying*

their position on the ring corresponding to y. It is easy to see now that

given Y , any collection Y = {yl1Y2, .... yn} such that y. lies in the interior

of (ijZj) specifies a flow f through each point in the network that follows

the (ordinary) shortest path tree corresponding to 6 and Y and separates on

each link (ij.,.) in the two opposite directions at the point yj. This flow

defines a length function 6 via the relation 6 y(t) = d[f y(t)] in the direction

of the flow, and 6 yields in the manner described earlier a shortest path

tree and a routing denoted by g(Y). It is easy to show (using the regularity

of Y ) that if Y is sufficiently close to Y then the (ordinary) shortest path

tree corresponding to 6 is the same as the one corresponding to Y and that

the elements of the routing g(Y) lie on the links (ijZJ ).

The algorithm described earlier can now be redefined a

(51) Yk+l = g(Yk)"

The definition is local within a sufficiently small neighborhood of Y and

is associated with the (ordinary) shortest path tree corresponding to Y

and the associated parameterization of the ring subnetworks containing the

links (ij,9j ).

Similarly as in the preceding section we say that an equilibrium

42

y is locally stable if there is a neighborhood of Y (defined in terms of

the parameterization of the rings associated with y as discussed earlier),

such that the sequence [g(ykuj generated by (50) is well defined and con-

verges to Y for every choice of Y0 within this neighborhood.

In order for Y to be locally stable it is sufficient that the

nxn matrix be defined and have all its eigenvalues within the unit

circle. The computation of ai(Y is straightforwqrd along the lines• 21Y

of Section 3. We first introduce some notation. For J -1,2,...,n

let + denote the set of points tE[yj,ej] on the jth ring, and R

denote the set of points tE[O,yj] on the same ring. NJote that for every

J,m-1,...,n the direction of flow on R and R+ (or R em) mustyj ,ej Ymaem ymm

coinside if these sets have intersection with positive Lebesgue measure.

This implies that at least one of the sets R+ nR+ and R+ n R"YJ,e I Ymem Yj yme m

is either empty or has Lebesgue measure zero. Similarly at least one of

the sets R e nRy and Ry fRr+ is either empty or has LAbesguey ,e y ,me m y ,ej ym,e

measure zero. The equations defining g(Y) can be written as

Pf AIF' d, = . rf+(Y,t)]dt, J -

tJ ) +- - * J "" -t. . .

Rgj (Y),e 1 (Y),e

By differentiation with respect to ym we obtain similarly as earlier at the

equilibrium Y

(g (Y , r(y*)

(51)o 2d (0) Jm'

where

43

d'[f+.(Y ,t)ldt(521 eJm N R+* +

yj,e. Ym,em

"4-. N- d'[fj(Y*,t)]dty ej Ym m

+R +d' [f-(Y ,t)ldt

R n R-*

+ * . .

y.,e.i yM ,e

d'[f.(Y*,t)ldt

and f+(Y ,t), f.(Y ,t) are the flows on the jth ring in the positive and

negative directions. In view of the preceding discussion, at least two of

the integrals in (52) are zero for every j and m.

Let R be the diagonal matrix having r(y.) as jth diagonal element,

and let @ be the nxn matrix having as elements the scalars 8 .m* Then we have

ag(y 1 8R.(Y 2d(O)

We can show that the matrix 0 is negative semidefinite. Indeed the matrix -Q

is the Gram matrix associated with the functions

XR +* e t d'.F(Y ,) - XR-* (t) d'!f.(Y ,t) j =n,

y] i yj,e.

where Xs is the characteristic function of a set S (X (t) = 1 if ttS, X(t) = 0

otherwise). By using the fact that R is diagonal it can be shown that the eiqpnvalu

X 'A of -g(Y )are real and nonpositive. Consider the spectral radius

n ay.

44

= max . . kThen the equilibrium Y is locally stable for

(53) I < 1,

and hence there exists a threshold level for d(O) above which the corrrsp;:,uing

equilibrium is stable. Similarly as in the preceding section, we can show

that if a fading memory scheme with decay factor is used to average the

effects of past routings the equilibrium Y is locally stable if

(54) 0 < 1+_ _

and there is a value of which optimizes the rate of convergence. It is

also possible to show that the other forms of averaging the effects of

several past routings improve the stability properties of the algorithm.

For the purpose of aiding the reader in understanding the method of

calculation of the matrix 9g(Y) we provide an example.3Y

Example: Consider the network shown in Figure 8 where node 4 is the

destination, and assume that the regular routing {y1 , Y2, Y 3 j shown in

an equilibrium. The figure shows also the chosen positive direction on

*the ring corresponding to each Y

f

45

+ +

4

Figure 8

We calculate the symmetric matrix 9 with elements e. given by (52).

The interval between any two nodes i and A is denoted [i,L]. The interval

between some y and a node I is denoted [yi,£]. We have

---- d'[f(Y ,t)] d--J* d'[f(Y ,t) dt

[Y2l1] U [1,3] U [3,41 [Yl ' 4 1

2 - 2 U d'[f + (Y*t)I dt d'lf2(Y*,t) dt22 [ 2,

2] [23 ( 2,l U[13

33 - ,d'If3(Y ,t] dt-J'* d'[f 3 (Y ,t)] dt

[Y3,2] U [2,3] U [3,4] [Y33 4J

1 - j d'[fC(Y*,t) dt

023'[+ *

(2,31

Li 46013 = - dfl(Y*,t)] dt

[3,4]

Acknowledgement

This investigation was conducted while the author participated in

a design study of the new ARPANET routing algorithm at Bolt, Beranuk, rl.,

Newman, Inc., Cambridge, Massachusetts. This study was supported by ARPA

under Contract MDA 903-78-C-0129. The collaboration and discussions with

John McQuillan, Ira Richer, and Eric Rosen had a substantial influence on

the ideas of this paper. The research was completed at the University

of Illinois, Urbana with support from Grant NSF-ENG 77-15949, and at the

Massachusetts Institute of Technology with support from Grant ONR-N00014-75-c-1183.

4-7

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: Proc. 1979 IEEE Cljnof!rt li e (,;i -'i' 1-1r -,.1 o, I 0

.; Ft. Lauderda, e, Fla Li , - i'", 9! -1,

WA6 9:45

D OY JNZc A F_', wL O"ITKItS Mi.

COK-tJN1FA- N LT ie .r

Department of flectfilal i r-,. -' .v,: . r s ter Zcienc

LaUorator-j I..,- :.!n - ar 2 S,, tsn

Abstract

This paper provides analysis ard computationa! . d--ia ti.e inate val P betaren t, and t in the

results relating to the dir.azsic behavior of shotte.t tlt 2

path routiny algorithms for store and forward co:"uri- cLzckw-se direction is def :!ed to be the setcation networks. A companion paper (1) focuses onnetworks with a single destination he ee wokt t t2 ,l if tl<t2

considers networks with multiple destindtivnf. - -

1. Introduction (t2,t 1 if ti>t2

This paper is a sequel to a companion paper Ill "which examines the dynamic behavior of shortest path for tI t2 we define Ptlt - t }-(t

routing algorithms for counication networks with a 2 1 2

single destination. We examine here the case where Notice tat we have for all tl.t 2 [O(l)

there are several destinations. Generally speaking, +the results obtained for single destir-tion network Rt t 2 2,

have extensions to the multiple destination case 1'2

although the analysis is considerably more complex.

Similarly as for single destination algocithus -' For each I - 1. N we a-- given a continuous function

find that the addition of a bias factor and the ri [0.I J 0,) .uch that r (0) - r1 ). We refer to

presence of an averaging mechanism have a stabilizin r, i-. N, a; the - : ,ensit-- function foreffect on dynamic behavior. Our rsults also suggest d I. Th nan; of ri is that for any two

that the presence of multiple destinations has a ... . ..da=ping effect on the dynamic behavior of the distinct points ti,t 2 an the ring the total traffic

algorithm. Similarly as in (1) we consider a network input 'Hat originates in R -* (Pt- ) a destinedoodel with a continuum of nodes and rely on methods tilt2 tiltof convergence analysis for discrete-time systems for x Iswith Euclidean state space. The possibilities for

extension of these results to finite node networks (eem limited. However, it appears that conclusons J rft)dt (f %(t)d)

on algorithmic behavior drawn fron analysis of tlhe t t2

continuous model remain at least qualitatively validfor finite node networks. The conjecture is supported w !cv Irtroeuc- n,tio.-.a of lenth ani distanceby results of ccmaptational experiments provided in al,,ng the rtrci. These notione will be considered inSection 3. Throughout the paper we ass une that the bth. t' c'Cikwiso and cointerclo,:kwise directions

&eader is famillia- with (1). which are also the negative and positive directions on[0.11. By a len-_,h fuactio:, we mean a function which

2. A Continuous Model of a Ring Network with assigns to each ;soint teO,lI two positive numfers 6*(t)

Multiple Destinationand (t), th- first associated with the positive

We consider a continuum of nodes arraned in a direction and tL- seond associated with the negative

ring and sending traffic to N points on the ring directi'n. Ths a length function can be identified

referred to as the destinations. A point is selected with the correspoonding pair (6+,6- )

.on the ring and is referred to as the orijin. Every

other point on the ring is Identified with Its In what fllo-' attention will be restricted to

tuclidean distance t from the origin in the ounter- lenqth fctinnis 5 - (6',6-

) for which 6" and 6- are

clockwise direction, where t is normalizel r- take piceise ccntl--. us on 10.1). Given such a length

values in the interval (0,i. The destinations are function wci the disrance AIt t ) Of any twoidentified with thcir di'ntances ,.. . from 12

t?e origin. We assume that p'nts *~ I t] I t ir tht. positive direct onki

0<X < x I .

Given any two distinct points ti , t2 on the rinq, th* ( J 6(t dt

interval R* t between t anJ t 2 i the ca'rnter- t L'2

clockwise direction is defined to b,! th- ,e ' . rti t. i-.rtarca P6 -(t IIt 2 of t1 end t2 In the

itt t r.;~t eirrelrnn is defkred b

t 2. . ) . (t)d (4)

It'l~l U Nt| if 1tlt

2I II I I

Given a length function (6+, C), it is clear that destint;.on points x 1,.. x. For every fixed tjf +-t)

there exists for every I a point z elol] such that and ,N)

z p' xi and f and f are continuous on (0l1) . Furtherore it

aa. 6( s .sn that for all I ad ye(0',IN

46|lx)- &,*(zi~xi). (5)

The point zi is unique except in the singular case -riz ) if teR t1) z

where zi 0 and z .1 both satisfy (5). Define ay i

Yi f dt, I - l,. n. (6)

ie

(0 if toWaNote that y1 is the counterclockwise Euclidean 32-Y1t)*

distance between xI and a, along the ring. Clearly we y( (1)

have 0<y 1 <l for all I. The vector Y - ( ' 1Y ; 1() if teI- tI -

defined by (5) and (6) is referred to as the shortestpath routing generated by (6+, 6-). It is easy to where z. is defined in terms of .y via (7).verify that Y has the property

Sizmilarly as in ill, we are given a continuously

X,* yl < X2 + Y2 < .<

X + YN diffe:e-tiable° monotonically increasing scalarfuncticn d of flow such that d(f)>G for all f>O.

more generally by a routing we mean an element (For si:licity we exclude the possibility d(O) . 0.e eIt was stown In [1) that unstable algorithmic behavior

of 10,1N

(the Cartesian product of N copies of 10,11). results in the single destination case if d(0) - 0).Given a routing Y - (yi" " YN

) consider for each I Given a routing Y we define the length function

for which y y' 1 the point z on the ring given by (6*, a-) corresponding to Y by

Sx, + Y, if X, + Y, <1 1 +

{x:-:y7} I t) - dlf*(Y.t)]. Ite0.11 - (14)

(715x, + Y, 1 if Xi I Yi >

I yt ~ ' tl t[,],(5

Define for each i - 1,...,N and tel0,1) the flow

t+(Yt) at t in the positive direction for destination We denote by q(Y) - [H1 (Y) .. (X) the shortest

I by pat-h routing generated by (. are interested in

0 if y -l, or yi1~ the algorithm

a(Yt)nd t 1k+l g(y )

te 8) where Y is a given initial ioating.If + i d 9,,311ndtasilt i'iwe say tht Ye is an .equilibrium routing if

Similarly define the flow f (Y.t) at t in the negative y* ; g(Y).

direction for destination I by Since g(Y) belongs to (0,1) for every routing T.it follows that an equilibrium Y* must belong to

0 if y, pl and teC N.( 2~ , ~ lo0l) Usk have the following proposition regardingt

' existeneC, uniqueness. and optimality of an

r (Odl if y1 il and te, *equil&ZrtrL. The proof i quite lengthy and has besef (It) - z1,t i i releqtad to Appendix A.

Proposition 1: There exists a unique equilibriumif y - 1. routinv ¥*. Furthelmore V* minimizes over all

Ye(0,1)"

the expression

Define also the total flows in the positive and nega- I I

tive direction at telo,13 by J(1) /ptf (Y.t)idt +f pt"f(Y.t)]dtN + 0-a

f+(Yt) - E f (Y.t) (10) where p is any function satisfying for all ri-I

f (Yt) I I f (Yt) (11)1-1 Ws tave assumed earlier that d(O)'O. If this

ass:tion is replsed by d(O)>O then Proposition INotice that for fixed T - (yl,...,y") the functions can still .e shown to hold excewpt for the uniqueness4(,.) and f (V,*) can be discontinuous only at the part. txistence of an equilibrium can be show by

usin ,AAutant's fixed point theorem (12). p. Vi) in

128~

place of Brower's theorem in the proof of App-,, . Then we can write the Jacobian (17) evaluated at ye kkThe proof of the optimality property of an eq ilP 'ugiven in Appendix A is also valid under the agr -tion - (22)d(O) > 0. (2

We now evaluate the Jacobian matrix We now establish that the matrix e is negative ssm i -

definite. Po- any set S denoted by X the characters-

g () ag (Y) tic fu;,ctio o! S, i.e. X (t) - I if teS, and X (-)is

if tos. It cm be seen that -e is the sum of twopositive se=44elinite matrices. The first matrix 1L

athe Graz matrx of the functions

i }¥ ~~ ~~X 10 ... i- .. I- . 3f

agN(Y

) 39 N(M in L1O,l1 (.31, p. 56), and the second matrix is the

ay 1 . . . .. ... ... Gram matrix of the functionsS3d(f+ (Y*,t)I

For any routing Ye(O.l)" define 9.(Y), i .. N, by X + (t)

the equation (cf. (7)]. i ziIi

Since every Gra matrix is positive seilAfnte it

i+9(Y) if x + 9 (Y)il follows l.st: 9 is negative secidefinite. Now the

ii ( Y )

" elgenvalues of -d .YD) are the same as thext+ 1 ( )-l if x 1 +g (P> eigenvalues of Ah! a . Dl DlGR1/ID which is

a negative semidfinite matrix. It follows that all

The equation defining gi(Y) is cf. (3) - (5). (14),(I)|eqgenvalucs I 1 ... of a re real end nonposi-,,

d[f-(Y,t)ldt - d(f+(Yt)ldt, tive. It is als possible to show that q(y h

9 (),xx i set of eigenvectors that form a basis for R", I.e. itis diagona--zablo. Let P ht the spectral radius of

Differentiation of this equation with respect to yjY

yields similarly as In M1 the following formulzo

-It lrtT.N vl* 411. 1T.kt(T)1l is locally stable if there exists -n~ghborhood JV of

I ¥* such that if yoe) then the sequence (Yk} generated

(18) by the algorithm 116) remains in N and converges to Y

Using Osto"swki's theorem (121. p. 300-301) and thewhere z4 and 24 are given by (7). At an equilLbirum fact that ha hs real gnvaluesnd i dia .a l"

(y1 ... y) we have izable we :a.n prove the following y .--. ....

Propositiz: 2: Am equilibrium Y - i i

, [ wi ... 4(locally stable if

S' '* ,'. '*'. , [1 it <(24)

cli n.. ,i .,&..u1 where 1 is defined by (23). urtherm if ( k t

(19) and Yk # " for all k there exists a norm " '

where for all i such that 1

#5 if .,Y i In I.1?z 3 i : - k-

Let N. ar-I L) De Lh. ),aisvna r (Z I- j + ;(df)

(d(f-(Y'.,2)I d df(Y*'/.. z

r , ertle -i .ie y - dIO) represents bia. It follows from thedia ooael ements. Let al, fl be the .y10etrl Nx, N prec.'- - a-ly-ts that a suff cetly high level of

vmatrix having elements bis w1,i -- oie a locally stahle equilibrium ard an

ar.j-r : q.nilax to the one of III shows that forsf,, :'y high level of bias the algorithm will con-

0.1 ~ ~ ~ ~ ~ ~ e j.. ' H 1 ~: .: . t-~ an #7vilIirt5 for every Initial routing I.

Le,,,-.

When there is a single destination (N-I) we have It follows from the fact <0(and (23) thatY * , . : .*Y . : -t 0o l w ar od th e d e n m i n t o i n<_ ( 1 9 ) a2 )

f(Yzi) - f+(Yz) -0 and the denominator in (19) the linearized system is stable ifequals 2a. When there are more than one destinations 1 + Bwe will usually have f(Y*,z) p' 0 and f+('Y,z.) 1 0 < 11

and the denominator in (19) can be significantly By copaxing this ineq ality with (24) we see that thelarger than 2i . Thus it is possible to have a stable fading =emory scheme improves significantly theequilibrium even if a is very small while this is not convergence properties of the algorithm. The value ofpossible when there is a single destination. Based on B for which an optimal rate of convergence is obtainedthis fact one is tempted to conclude that the presence is the one that minimizes max fiB + (1-B) ll) -of more than one destination tends to have a benefi- i

cial effect on the dynamic behavior of the algorithm. Siiiarly one can obtain extensions of other resultsThis conjecture is supported by the results of compu- obtained Ln EI in connection with averagingtational experiments, but we have been unable to aigorithos for the single destination case.formulate it and establish it mathematically. 3. Cocoutational Results

As in (1), it is possible to define and analyze

an algorithm where the bias a depends on the current We experimented with a 30-node ring network androuting Y Similar results as those of (11 can be consiee- d a synchronous and an asynchronous algorithm

with evenly spaced delay broadcasts. A fading mmoryobtained, so we ormit the details. scheme was used to average delays corresponding to

the present and past routings. In this scheme "delays'Basing the Routing Decision on More than one Past are computed at each iteration by means of the formulaRoutings

It was seen in Ill that the dynamic behavior ofthe algorithm for the single destination case can be [New Delay of Link (i)I -

improved by employing some form of averaging of the -B[Old Delay of Link (ii)I +effects of several past routings. This analysis andthe corresponding conclusions can be generalized to + (l-0)(0.05) (Current flow of (i,U)|the multiple destination case.

As an example consider the case where the length (25)function Ik, 6 ) used in generating Yk+l depends on The scalar B is referred to as the decay factor and

all previous routings Yk' Ykl via a fading takes values in the interval 10.1). Large values ofkeory s he e OB im pjy a greater degree of averagin g with delays

memory scheme of the form corresponding to past routings. The length of link

(i,.I) used in the shortest path computation was taken6k(t) - B06klt) + (1-O)dIf(Yk't)], vte[ol to be I

+ (t) - 06- (t) + Cl-O)dlf (Yk 0) * vte[,l) [Length of Link (,)1- Bias + (New Delay Link (,fJlk k-I k' The bias depends on the current routing (c.f. (11)

and was taken to be "--

where Be(0,1) is the decay factor. Then the linearized

system corresponding to an equilibrium Y* is given Bias - 0.02 x (Sum of most recentl- "reported delayiby (cf.(lj over all links).

,(~l_ )j_ + 82 In the synchronous algorithm ll nodes "reWort"

k+l ay k+ Yk-2 ] their link delays simultaneously at each iterationWrite and all of these delays are used in the shortest path

computation that determines the new routing.

- YT + BY.. + In the asynchronous algorithm nodes comipte their... a delays at every iteration by using equation (25).

Then we have for all k mowever they "report" their link delays only everyC 30th iteration, with node I reporting at the let

Yk+l a(-0) Yk+0

(1 -0)

2 k iteration, node 2 at the 2nd iteration and so on. Thasfor exan=ple at the 31st iteration node I will report

z l i YkB " his link delays and the new routing will be competedSk0 Ii I k + kon the basis of the delays reported by node 2 at the

This system is stable if all sigenvalues of the 2nd iteration, by node 3 at the third iteration, and

matri so on. This procedure is patterned after the asyn-

1 -8 01-B) I (Y The traffic inputs for destination 30 are given at

matrix chroncus operation of he algorithm in the MAnT.

3Y the bottom of tables 1-3. The traffic sent by every

node to eath of the other destinations 1.2.29 was

I I taken to be the same and equal to A. We consider two

where I is the NOH identity matrix, lie within the values A - 0.05 and A - 1. In all runs the initialunit circle. Since this matrix has rank H it has N routirg for destination i was taken to be (I + 15)unitcirle. inc thi marix as ank ith N (r-odulo 3 1. We show in" tables I - 3 the generatedeigenvalues equal to zero. Its remaining N eigen- Wvalues can be shown to equal the eigenvalues of the sequences of successive routings for destination 30,

(.e. the s.quences of the numerically largest nodematrx BI + (1-0) 3 which are '(B)A, roIti' ; ressages to destination 30 in the Clockwise

a d rect o). The sequences of routings for the otherI,...,N where A i is the ith eigenvalue of dest rsatlons are not shown. They exhibited very

130

similar behavior. An entry a/b for the a:3' - xoncus node netvorks. Furtherore, as the level of X of traffic

algorithm means that during a 30-iteration cycle the input to the destinations I through 29 is increasedandnminimum and l&simum routirgs are a and b. The results te contribution of all destinations to the total traf-indicate that all the conclusions reached regarding fic is rmre nearly equal, th. dynamic behavior of the

the effects of bias and averaging are valid for finite algo-it-s Is i~proved.

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Acknowledgement:[1 IThis investigation was conducted in part at Solt n.inmize oPtf+(t)ldt + p[f'(t)lJdt

Beranek, and Newman, Inc., Cambridge, Mass. and support- Ned by ARPA Contract MDA 903-78-C-0129 and in part at subject to I riJT)uI(T)dT - f+(t) - 0the University of Illinois, Urbana, Ill., and supported i- xi t 'by Grant NSF-ENG-77-15949. The comments of JohnMcQuillan, Ira Richer and Eric Rosen, and the assist- &.a. te(0,l1.ance of Eli Gafni with the computational experimentsare gratefully acknowledged. N 4 r )u T)dT - t 0.

References: a~.tc[0,1|

(11 D.P. Bertskas, "Dynamic Models of Shortest PathRouting Algorithms for Communication Networks', +Laboratory for Information and Decision Systems 0 <u ut(t), u 1 (t)Working Paper, Massachusetts Institute of Techno-logy, Cambridge, Mass., June 1979. a.a. te[o,lj,

121 J.M. Ortega and W.C. Pheinboldt, Iterative f4* , "~ u- eL.2 m(O., i .Solution of Nonlinear Equations in SeveralVariables, Academic Press, N.Y., 1970. where a.a.te[O,l means for almost all teto.i1 with

respect to Lebesgue measure.

(3) D.G. Luenberger, Optimization by Vector SpaceMethods, J. Wiley, N.Y., 1969. To each routing Y - CyL. N

) with correspond-

ing vector Z - (z,.- . ZN) defined by (7) we can

associate the following feasible solution of problem(A.2)

Appendix A: Proof of Proposition 1 f+(t) - f (Yt). f-(t) - f (Yt) (A.3)

The function g: (0,11 N . 10,11" is clearly continuous

so by Brower's fixed theorem ((21, p. 161) it has afixed point Y* which is an equilibrium. 0 if tlRi

n order to show that Y In unique and minimizes uIt) = Mover all Yetom I if e

J (Y) - p+tf+(Yt)ldt +f1 pt f(Yt)ldt (A.l) zi'xi

we consider an associated convex programing problem. 0 if tAi"

Let L 10,11 be the space of (equivalent classes of) u1 (t) I i*xi

square integrable function on 10,1] with norm I if tea1 2 1/2

Ilfil- (fO If(t)32 dt)l/dt (A.4)

The Cartesian product LV of n copies of L2(0,1. where The corresponding value of the objective of problem

(A.2) Is J(Y). The fact that an equilibrium routingn is any positive Integer, will have the norm i . f he f olloing lm ot

n 2 1/2minimizes J(Y) follows from the following laMas

II[ .)tI ) 12

i Lema A.1: If Y is an equilibrium routing them the

functions f . f ui.u;. l-.... o ,,s9 nd-With this norm L(O,l is a Hilbert space with inner ing to Y, via equations (A.3) and (A.4) are an optimalproduct solution of problem (A.2).

<(f. f )(g .. gn > - M dt , Proof, Consider the functions dlf IT .. )).

fdf all d....Ye)L 2 (0,l. By the sufficiency theorem of

for all If 1 . ,n L:(Ol3,(g.. . gn)eL[0,1. Any (33, p. 220 the result will follow if we show that

element (g. ..... )eP10 ,1) defines a bounded linea f*+ "s

functional on L3(0,11 by means of the preceding equa- problem

tion, and all bounded linear functionals on L [,1 pm2

can be defined this way.

We consider the following convex programming

problem in j2N2

(0 ,11 which fells within the frame-2

work given in Luenberger ((33, Chapter 8)

132

minimize ptff(t)I + pit-() that (A.7) holds for all te[O,li. This together withthe assption d(O) >0 implies that , T2 . Q.E.D.N

+

dif-(Y IM I) l,[ I: M :. () 4T-f'lt)) atI Ixi.t

--i (atI + O (t uj(t) +u;)4(t) I

subject to O<u+ (t), ,t), u+(t) + u -

e.a.te 0 l, i - 1 ... , .

CA.S)

Define

bCtM - f d f+ CT* -t) dT

D: (t) - f -tAx

A straightforward calculation using integration byparts shows that problem (A.5) can be written as

minimizeO{pf+(t)] + ptf t)]

- d(f+(Y*,t)]f*Ct) - daf-C.t)3f t)

subject to O<u+(t), O_ t u * u;t) - 1.

a.a.te[O,l), i-l,....,H " A-

UsLna the fact that - d, It is easy to see tbat

I U, UL I - 1 ,..°N are an optimalsolution to this problem. Q.E.D.

It remains to show uniqueness of the equilibrium.Since d(f) > 0 for all t, all d is monotonicallyincreasing, the function p is strictly convex. LAt Fbe the convex set of all(f ,f-)eL2(0,11 for which

there exist f+. C. ul. u1 I - 1....,4 which are

feasible for problem (A.2). Problem (A.2) can bewritten as

minimszefJ (pif+ t) + ptf-lt)3)dt (A-6)

subject to (ff')e•.

If Y; and Y; are to equilibria, then

tf*(¥;" f (Yl*," ) mnd'f (Y;,' , " ;,',]

are both optimal solutions of problem (A.6). Using theetict convexity of p, it follows that except for tin a aet having Lebesgue measure zero

+ -f;.t, ,.,;.t),. ,-(,.,t) ,-Y;,t).

+ *(A.7)Since f(ty;,.), f (Y2 .'.), f CT f1 (,.', are

continuous overywhere except at x1 . xN, it follows

133